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Yes the rich would probably pay more even with getting their basic income. The point of giving it to everyone is to reduce the administrative cost of figuring out who qualifies, and to get rid of "welfare cliffs" that give poor people a disincentive to earn more for fear of losing assistance programs.

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Essentially, it avoids duplicating income-based computations across tax system and multiple benefit programs, and concentrates then in the tax system (once you have a mature UBI that has displaced other benefit programs.)

In this sense, it can be seen as an application of the DRY principle.

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